Download Standards for Mathematical Practice

Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Standards for Mathematical Practice
The Mathematical Practice standards describe expertise that teachers should seek to develop in students. The process identified within in these standards will
be a part of everyday classroom instruction.
MP 1. Make sense of problems and persevere in solving them.
MP 2. Reason abstractly and quantitatively.
MP 3. Construct viable arguments and critique the reasoning of others.
MP 4. Model with mathematics.
MP 5. Use appropriate tools strategically.
MP 6. Attend to precision.
MP 7. Look for and make use of structure.
MP 8. Look for and express regularity in repeated reasoning.
Fluency Recommendations:
G‐SRT.B.5 Fluency with the triangle congruence and similarity criteria
G‐GPE.B.4, 5, 7 Fluency with the use of coordinates
C‐CO.D.12 Fluency with the use of construction tools
Page 1
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Page 2
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Page 3
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Unit I – Fundamentals of Geometry with Angles and Lines
Description: This unit gives students the foundational tools for developing viable geometric arguments using relationships students studied in middle school
related to lines, transversals, and special angles associated with them. Students learn how to combine true statements within a mathematical system to
deductively prove other statements. Students should begin to see the structure of a mathematical system as they make conjectures and then prove statements
involving lines and angles.
Domain
Tennessee State Standards
Experiment with G-CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of
Transformations point, line, distance along a line, and distance around a circular arc.
in the Plane
(CO.A)
G-CO.C.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines,
Prove Geometric
alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are
Theorems (CO.C)
exactly those equidistant from the segment’s endpoints.
Use Coordinates G-GPE B.4 Use coordinates to prove simple geometric theorems algebraically.
to Prove Simple G-GPE.B.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of
Geometric
a line parallel or perpendicular to a given line that passes through a given point).
Theorems
Algebraically
(GPE.B)
Corresponding Baseline Literacy Module
http://www.jmcss.org/pages/JMCSS/Departments/Curriculum/Resources/BLM/HS/Geometry/Module1
Page 4
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Unit II – Geometric Transformations
Description: In high school, students formalize much of the geometric exploration from middle school. In this unit, students develop rigorous definitions of
three familiar congruence transformations: reflections, translations, and rotations and use these transformations to discover and prove geometric properties.
Throughout the course, students will use transformations as a tool to analyze and describe relationships between geometric figures.
Domain
Tennessee State Standards
G-CO.A.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of
point, line, distance along a line, and distance around a circular arc. (In this unit, students will extend the fundamental terms that will be
used throughout the course.)
G-CO.A.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions
Experiment with that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those
Transformations that do not (e.g., translation versus horizontal stretch)
in the Plane
G-CO.A.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
(CO.A)
G-CO.A.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line
segments
G-CO.A.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing
paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Understand
G-CO.B.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given
Congruence in figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
Terms of Rigid
Motions (CO.B)
Page 5
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Unit III – Triangles
Description: This unit explores basic theorems and conjectures about triangles, including the triangle inequality conjecture, the Triangle Sum Theorem, and
theorems regarding centers of a triangle. Students explored some of these relationships in middle school but will build on their work in unit 2 with deductive
reasoning and proof related to triangles in this unit. Students make and verify conjectures related to isosceles triangles and explore physical properties of the
centroid of a triangle. In this unit, students also learn basic construction techniques and use these as they explore triangle properties. Throughout this unit,
students will use the precise definitions developed in G-CO.A.1.
Domain
Prove
Geometric
Theorems
(CO.C)
Tennessee State Standards
G-CO.C.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles
triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians
of a triangle meet at a point.
G-CO.D.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices,
Make
paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing
Geometric
perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not
Constructions
on the line.
(CO.D)
G-CO.D.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Understand G-C A.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
and Apply
Theorems
about Circles
(C.A)
Apply
G-MG A.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a
Geometric
cylinder).★
Concepts in
Modeling
Situations
(MG.A)
Page 6
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Unit IV – Triangle Congruence
Description: This unit builds on students’ work with transformations in Unit II and properties of triangles in Unit III to formalize the definition of congruent
triangles. Students reason to identify criteria for triangle congruence and use precise notation to describe the correspondence in congruent triangles.
Domain
Understand
Congruence in
Terms of Rigid
Motions
(CO.B)
Tennessee State Standards
G-CO.B.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure;
given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
G-CO.B.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding
pairs of sides and corresponding pairs of angles are congruent.
G-CO.B.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid
motions.
Page 7
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Unit V – Similarity Transformations
Description: This unit moves away from rigid motion and focuses on dilations and similarity. Students prove theorems involving similarity and apply dilations
and similarity to model situations in the real world.
Domain
Prove Geometric
Theorems (CO.C)
Understand Similarity
in Terms of Similarity
Transformations
(SRT.A)
Prove Theorems
Involving Similarity
(SRT.B)
Use Coordinates to
Prove Simple
Geometric Theorems
Algebraically (GPE.B)
Apply Geometric
Concepts in Modeling
Situations (MG.A)
Tennessee State Standards
G-CO.C.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of
isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the
length; the medians of a triangle meet at a point.
G-SRT A.1. Verify experimentally the properties of dilations given by a center and a scale factor:
a.) A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the
center unchanged.
The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G-SRT.A.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar;
explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and
the proportionality of all corresponding pairs of sides.
G-SRT.A.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
G-SRT.B.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two
proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G-SRT.B.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G-GPE B.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G-MG A.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or
minimize cost; working with typographic grid systems based on ratios).★
Corresponding Baseline Literacy Module
http://www.jmcss.org/pages/JMCSS/Departments/Curriculum/Resources/BLM/HS/Geometry/Module3
Page 8
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Unit VI – Right Triangle Relationships and Trigonometry
Description: This unit extends the idea of similarity to indirect measurements. Students develop properties of special right triangles, and use properties of
similar triangles to develop trigonometric ratios. Students apply these ideas as they model real-world situations and solve problems involving unknown side
lengths and angle measures.
Domain
Prove Theorems
Involving
Similarity
(SRT.B)
Define
Trigonometric
Ratios and Solve
Problems
Involving Right
Triangles (SRT.C)
Use Coordinates
to Prove Simple
Geometric
Theorems
Algebraically
(GPE.B)
Apply Geometric
Concepts in
Modeling
Situations
(MG.A)
Tennessee State Standards
G-SRT.B.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G-SRT.C.6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of
trigonometric ratios for acute angles.
G-SRT.C.7. Explain and use the relationship between the sine and cosine of complementary angles.
G-SRT.C.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
G-GPE.B.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★
G-MG A.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder).★
G-MG.A.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or
minimize cost; working with typographic grid systems based on ratios).★
Page 9
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Unit VII – Quadrilaterals
Description: Prior units of this course have focused on triangles. This unit extends that work to the study of quadrilaterals. Students use triangle congruence as
they prove theorems about parallelograms. This unit also provides an opportunity for students to become proficient with coordinate proofs.
Domain
Prove Geometric
Theorems (CO.C)
Use Coordinates
to Prove Simple
Geometric
Theorems
Algebraically
(GPE.B)
Make Geometric
Constructions
(CO.D)
Tennessee State Standards
G-CO.C.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the
diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
G-GPE B.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four
given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and
containing the point (0, 2).
G-CO.D.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Corresponding Baseline Literacy Module
http://www.jmcss.org/pages/JMCSS/Departments/Curriculum/Resources/BLM/HS/Geometry/Module2
Page 10
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Unit VIII – Understanding and Modeling with Two- and Three-Dimensional Figures
Description: This unit explores three-dimensional geometry including representations of real-world situations with three-dimensional objects and calculating
volume. Students make connections between two-dimensional and three-dimensional representations of objects. Students culminate the course with
modeling problems involving three-dimensional objects, allowing them again to integrate their knowledge and apply complex geometric reasoning.
Domain
Explain
Volume
Formulas and
use them to
Solve
Problems
(GMD.A)
Visualize
Relationships
between TwoDimensional
and ThreeDimensional
Objects
(GMD.B)
Apply
Geometric
Concepts in
Modeling
Situations
(MG.A)
Tennessee State Standards
G-GMD A.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and
cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
G-GMD.A.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★
G-GMD.B.4. Identify the shapes of two-dimensional cross-sections of three- dimensional objects, and identify three-dimensional objects
generated by rotations of two-dimensional objects.
G-MG.A.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder).★
G-MG.A.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★
G-MG.A.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or
minimize cost; working with typographic grid systems based on ratios).★
Page 11
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Unit IX – Circles
Description: This unit explores properties of circles. Students draw on geometric relationships involving lines, angles, triangles and quadrilaterals as they derive
the equation of a circle and explore properties of chords, arcs, and angles on circles.
Domain
Tennessee State Standards
Make Geometric G-CO.D.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Constructions
(CO.D)
G-C.A.1. Prove that all circles are similar.
Understand and G-C.A.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed,
Apply Theorems and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the
About Circles radius intersects the circle.
(C.A)
G-C.A.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a
circle.
Find Arc Lengths G-C.B.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian
and Areas of
measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Sectors of Circles
(C.B)
Translate
G-GPE A.1.Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center
Between the
and radius of a circle given by an equation.
Geometric
Description and
the Equation for
a Conic Section
(GPE.A)
Use Coordinates G-GPE.B.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four
to Prove Simple given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and
Geometric
containing the point (0, 2).
Theorems
Algebraically
(GPE.B)
Page 12
Geometry Pacing & Assessment Guide
JMCSS DEPARTMENT OF CURRICULUM & INSTRUCTION
Explain Volume
Formulas and
Use them to
Solve Problems
(GMD.A)
Apply Geometric
Concepts in
Modeling
Situations
(MG.A)
G-GMD A.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid,
and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
G-MG A.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder).★
Page 13